Abstract
In this paper, we study the finite dimensional modules over indefinite Kac–Moody Lie algebras. We prove that any Kac–Moody Lie algebra with indecomposable indefinite Cartan matrix has no non-trivial finite dimensional simple module. This result would be indispensable for researching finite dimensional modules over GIM Lie algebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berman S., On generators and relations for certain involutory subalgebras of Kac–Moody Lie algebras. Comm. Algebra, 1989, 17(12): 3165–3185
Carbone L., Chung S., Cobbs L., McRae R., Nandi D., Naqvi Y., Penta D., Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits. J. Phys. A, 2010, 43(15): 155209, 30 pp.
Frenkel I., Lepowsky J., Meurman A., Vertex Operator Algebras and the Monster. Pure Appl. Math., Vol. 134, Boston, MA: Academic Press, Inc., 1988
Gao Y., Xia L.M., Finite-dimensional representations for a class of generalized intersection matrix Lie algebras. Comm. Algebra, 2016, 44(11): 4794–4810
Kac V.G., Graded Lie algebras and symmetric spaces. Funkcional. Anal. i Priložen, 1968, 2(2): 93–94 (in Russian)
Kac V.G., Infinite-dimensional Lie Algebras, 3rd Edition. Cambridge: Cambridge University Press, 1990
Kostant B., On Whittaker vectors and representation theory. Invent. Math., 1978, 48(2): 101–184
Li W.L., Classification of generalized Cartan matrices of hyperbolic type. Chinese Ann. Math. Ser. B, 1988, 9(1): 68–77
Mazorchuk V., Lectures on \({\mathfrak{s}\mathfrak{l}_2}(\mathbb{C})\) (ℂ)-modules. London: Imperial College Press, 2010
Moody R.V., Lie algebras associated with generalized Cartan matrices. Bull. Amer. Math. Soc., 1967, 73: 217–221
Nilsson J., \({\cal U}(\mathfrak{h})\)-free modules and coherent families. J. Pure Appl. Algebra, 2016, 220(4): 1475–1488
Saclioğlu C., Dynkin diagrams for hyperbolic Kac-Moody algebras. J. Phys. A, 1989, 22(18): 3753–3769
Slodowy P., Beyond Kac-Moody algebras, and inside. In: Lie Algebras and Related Topics, CMS Conf. Proc., Vol. 5, Providence, RI: AMS, 1986, 361–371
Tan H.J., Zhao K.M., Irreducible modules over Witt algebras Wn and over \({\mathfrak{s}\mathfrak{l}_{n + 1}}(\mathbb{C})\). Algebr. Represent. Theory, 2018, 21(4): 787–806
Wan Z.X., Introduction to Kac–Moody Algebra. Teaneck, NJ: World Scientific Publishing Co., Inc., 1991
Acknowledgements
This work is partially supported by NSFC (Nos. 11871249, 12171555, 11801394) and NSF of Jiangsu University (No. 5501190011).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The authors declare no conflict of interest.
Rights and permissions
About this article
Cite this article
Xia, L., Hu, H. & Tan, Y. Finite Dimensional Modules over Indefinite Kac–Moody Lie Algebras. Front. Math 19, 161–170 (2024). https://doi.org/10.1007/s11464-022-0072-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-022-0072-8